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Step-By-Step Optimization With Excel Solver              The Excel Statistical Master
Copyright ©2011 http://excelmasterseries.com/New_Manuals.php
Page 170
Sensitivity Report
The Sensitivity Report is also made available when the Solver finds a
Globally or Locally Optimal solution and no Integer Constraints (Integer,
Binary, Alldifferent) were used. Part 2 of the Answer Report Shows that
the variables were Continuous and not Integers. The Simplex LP method
solves linear problems to globally optimal solutions.
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Step-By-Step Optimization With Excel Solver              The Excel Statistical Master
Copyright ©2011 http://excelmasterseries.com/New_Manuals.php
Page 171
Optimal Investment Selection Example
Maximizing Investment Return Through Optimal
Investment Selection Using the Binary Constraint
The Solver can be used to maximize return when selecting investments
with a limited amount of investment capital. Projected cash flows for
each investment are Solver inputs. Binary Decision Variables
determining whether or not an investment will be made. The Solver
calculates the Net Present Value of each possible combination of
investments and determines the investment combination that maximizes
Net Present Value (NPV) of invested cash flows.
The Problem
A venture capitalist with limited funds can make annual
investments over the next 2 years. He has 6 investment
opportunites to choose from. 3 of the investment opportunities will
occur at the start of year 1. The other 3 investment opportunities
will occur at the start of year 2. He doesn’t have to invest anything
if he chooses not to. If he does choose to invest in any opportunity,
he must invest upfront 100% of start-up capital required by that
investment. The venture capitalist has only enough capital to make
2 investments during any one year.
The projected annual cash flows for each investment are shown
below.  These projected cash flows include the upfront investment
that the venture capitalist would have to make.
The goal is to select the investments that would maximize Net
Present Value at year 0 of all cash flows of all selected investments.
A discount rate of 25% will be used because the investments are
considered risky.
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Step-By-Step Optimization With Excel Solver              The Excel Statistical Master
Copyright ©2011 http://excelmasterseries.com/New_Manuals.php
Page 172
The projected annual cash flows for each investment, including all
upfront investments, are shown as follows:
Problem Solving Steps
Step 1 – Determine the Objective
In this case, the objective is to maximize the Net Present Value at Year 0
of all cash flows of all selected investments. The cell in which the Net
Present Value at Year 0 is calculated is the Objective Cell.
Step 2 – Determine the Decision Variables
We are trying to select the investments which will produce the highest
Net Present Value of all cash flows at Year 0. The Decision Variables
are binary variables (taking values of 1 or 0) which indicate whether an
investment opportunity was chosen.
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Step-By-Step Optimization With Excel Solver              The Excel Statistical Master
Copyright ©2011 http://excelmasterseries.com/New_Manuals.php
Page 173
Step 3 – Build the Excel Equations That Combine
the Objective With All Decision Variables
On the following pages are expanded views of the left and right sides of
the preceding Excel model for better clarity.
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Step-By-Step Optimization With Excel Solver              The Excel Statistical Master
Copyright ©2011 http://excelmasterseries.com/New_Manuals.php
Page 174
Shown as follows is the left side of the Excel model. The green Decision
Variables (C17 to E17, F18 to H18) are binary variables occurring in
Years 1 and 2. The light blue Constraints cell (C24 and F24) limit the
maximum number of investments that can be made during Year 1 and
Year 2.
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Step-By-Step Optimization With Excel Solver              The Excel Statistical Master
Copyright ©2011 http://excelmasterseries.com/New_Manuals.php
Page 175
The right side of the model shown as follows contains 1) the combined
annual cash flows for all selected investments (Column I), 2) the Present
Value at Year 0 of each combined annual cash flow (Column J), 3) and
the yellow Objective cell (K17) containing the Net Present Value at Year
0 of all combined annual cash flows (the sum of the Year 0 Present
Values). Each year’s combined cash flows (I17 to I22) were copied from
cells I31 to I36. The Excel formula for calculating the Present Values at
Year 0 in column J is also shown.
Step-By-Step Optimization With Excel Solver              The Excel Statistical Master
Copyright ©2011 http://excelmasterseries.com/New_Manuals.php
Page 176
Calculation of Net Present Value (NPV) of All Cash
Flows At Year 0
The Net Present value (NPV) of all cash flows at year 0 equals the sum
of the Present Value (PV) of each total annual cash flow at Year 0.
The Present Value at Year 0 of a cash flow that occurs in Year t is:
PV
t=0
= C
t
(1 + i)-t
C
t
= Cash flow C that occurs in year t
I = Discount rate – We are discounting all cash flows back using a
discount rate of 25% because the investment is considered risky, as
most venture capital investments are. The Discount Rate is user-
controlled.
For example, the Present Value at Year 0 of the Year 4 Cash flow,
\$115,000, would be calculated as follows:
PV
t=0
= C
t
(1 + i)-t
PV
t=0
= (\$115,000)(1 + 0.25)-4 = \$47,104